Balance constants for Coxeter groups

The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.Comment: 27 page

The sum-capture problem for abelian groups

Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq G$ be a random set of size $|G|^\alpha$. We let $$\mu(A) = \max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|.$$ The issue is to determine upper bounds on $\mu(A)$ that hold with high probability over the random choice of $A$. Mennink and Preneel \cite{BM} conjecture that $\mu(A)$ should be close to $|A|$ (up to possible logarithmic factors in $|G|$) for $\alpha \leq 1/2$ and that $\mu(A)$ should not much exceed $|A|^{3/2}$ for $\alpha \leq 2/3$. We prove the second half of this conjecture by showing that $$\mu(A) \leq |A|^3/|G| + 4|A|^{3/2}\ln(|G|)^{1/2}$$ with high probability, for all $0 < \alpha < 1$. We note that $3\alpha - 1 \leq (3/2)\alpha$ for $\alpha \leq 2/3$. In previous work, Alon et al$.$ have shown that $\mu(A) \leq O(1)|A|^3/|G|$ with high probability for $\alpha \geq 2/3$ while Kiltz, Pietrzak and Szegedy show that $\mu(A) \leq |A|^{1 + 2\alpha}$ with high probability for $\alpha \leq 1/4$. Current bounds on $\mu(A)$ are essentially sharp for the range $2/3 \leq \alpha \leq 1$. Finding better bounds remains an open problem for the range $0 < \alpha < 2/3$ and especially for the range $1/4 < \alpha < 2/3$ in which the bound of Kiltz et al$.$ doesn't improve on the bound given in this paper (even if that bound applied). Moreover the conjecture of Mennink and Preneel for $\alpha \leq 1/2$ remains open

The variance conjecture on projections of the cube

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in S^{n-1}}{\mathbb E}_\mu\langle x,\theta\rangle^2{\mathbb E}_\mu|x|^2,$$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces